where \(\mathbf {K}_{mm}\)  is the matrix of covariances between the inducing inputs. This prior follows from the assumption that the inducing outputs also behave like the latent variable \(f\)  which has a Gaussian prior. Therefore, \(\mathbf {u}\)  and \(\mathbf {f}\)  have a joint Gaussian distribution [1]} given by
\(\begin{split}p(\mathbf {f}, \mathbf {u}) =& p(\mathbf {f|\mathbf {u}})p(\mathbf {u}) \\=& \mathcal {N}(\mathbf {K}_{nm}\mathbf {K}^{-1}_{mm}\mathbf {u}, \mathbf {K}_{nn} - \mathbf {Q}_{nn}) \times \mathcal {N}(\mathbf {0}, \mathbf {K}_{mm}),\end{split}\) 
